Best Known (13, 25, s)-Nets in Base 27
(13, 25, 123)-Net over F27 — Constructive and digital
Digital (13, 25, 123)-net over F27, using
- net defined by OOA [i] based on linear OOA(2725, 123, F27, 12, 12) (dual of [(123, 12), 1451, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(2725, 738, F27, 12) (dual of [738, 713, 13]-code), using
- construction XX applied to C1 = C([725,7]), C2 = C([0,8]), C3 = C1 + C2 = C([0,7]), and C∩ = C1 ∩ C2 = C([725,8]) [i] based on
- linear OA(2721, 728, F27, 11) (dual of [728, 707, 12]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−3,−2,…,7}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2717, 728, F27, 9) (dual of [728, 711, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(2723, 728, F27, 12) (dual of [728, 705, 13]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−3,−2,…,8}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2715, 728, F27, 8) (dual of [728, 713, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(272, 8, F27, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([725,7]), C2 = C([0,8]), C3 = C1 + C2 = C([0,7]), and C∩ = C1 ∩ C2 = C([725,8]) [i] based on
- OA 6-folding and stacking [i] based on linear OA(2725, 738, F27, 12) (dual of [738, 713, 13]-code), using
(13, 25, 164)-Net in Base 27 — Constructive
(13, 25, 164)-net in base 27, using
- 1 times m-reduction [i] based on (13, 26, 164)-net in base 27, using
- (u, u+v)-construction [i] based on
- (2, 8, 82)-net in base 27, using
- base change [i] based on digital (0, 6, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- base change [i] based on digital (0, 6, 82)-net over F81, using
- (5, 18, 82)-net in base 27, using
- 2 times m-reduction [i] based on (5, 20, 82)-net in base 27, using
- base change [i] based on digital (0, 15, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81 (see above)
- base change [i] based on digital (0, 15, 82)-net over F81, using
- 2 times m-reduction [i] based on (5, 20, 82)-net in base 27, using
- (2, 8, 82)-net in base 27, using
- (u, u+v)-construction [i] based on
(13, 25, 470)-Net over F27 — Digital
Digital (13, 25, 470)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2725, 470, F27, 12) (dual of [470, 445, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(2725, 737, F27, 12) (dual of [737, 712, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(2723, 729, F27, 12) (dual of [729, 706, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2717, 729, F27, 9) (dual of [729, 712, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(272, 8, F27, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(2725, 737, F27, 12) (dual of [737, 712, 13]-code), using
(13, 25, 105987)-Net in Base 27 — Upper bound on s
There is no (13, 25, 105988)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 608295 928706 555952 726926 847727 418841 > 2725 [i]